An option would be Mary, taken after King George VI’s mother, Mary of Teck, whose mother was Princess Mary Adelaide of Cambridge, and after Queen Elizabeth II’s third name, Elizabeth Alexandra Mary.
Another possible name is Alexandra, Queen Elizabeth II’s second name, or King George V’s mother, Alexandra of Denmark.
Cecilia is also a possibility, inspired by the name of the Queen Mother’s mother, Cecilia Bowes-Lyon, Countess of Strathmore and Kinghorne.
Victoria comes from Queen Victoria, the mother of King Edward VII. Also Queen Victoria’s mother was Princess Victoria of Saxe-Coburg-Saalfeld.
Diana is obviously taken after Princess Diana, Elizabeth after Queen Elizabeth II’s first name, and Frances after Princess Diana’s second name and Frances Dora Smith, the Queen Mother’s paternal grandmother.
Charlotte is a name that would pay tribute to Princess Charlotte of Wales, the daughter of King George IV. Her mother’s name was Charlotte of Mecklenburg-Strelitz.
Louise is taken after Louise of Great Britain, her parents were George II of Great Britain and Caroline of Ansbach.
And finally Alice after Princess Alice of Hesse, whose mother was Queen Victoria and Princess Alice of Battenberg, the mother of Prince Philip, Duke of Edinburgh.